Properties

General

Proposition

If XX has the property that every face of every non-degenerate simplex is again non-degenerate, then the inclusion of the category of non-degenerate simplices (Δ↓X)nondeg↪(Δ↓X)(\Delta \downarrow X)_{nondeg} \hookrightarrow (\Delta \downarrow X) has a left adjoint and is hence a reflective subcategory.

Theorem

Proof

An nn-simplex of N(Δ↓X)N(\Delta\downarrow X) is determined by a string of n+1n+1 composable morphisms

Δkn→…→Δk0 \Delta^{k_n} \to \dots\to \Delta^{k_0}

along with a map Δk0→X\Delta^{k_0} \to X, i.e. an element of Xk0X_{k_0} Thus, each the functor X↦N(Δ↓X)nX\mapsto N(\Delta\downarrow X)_n from SSet→SetSSet \to Set is a coproduct of a family of “evaluation” functors. Since evaluation preserve colimits, coproducts commute with colimits, and colimits in SSetSSet are levelwise, the statement follows.

Therefore, the simplicial set N(Δ↓X)N(\Delta\downarrow X) itself can be computed as a colimit over the category (Δ↓X)(\Delta\downarrow X) of the simplicial sets N(Δ↓Δn)N(\Delta\downarrow \Delta^n).